Dependence relations in computably rigid computable vector spaces

We construct a computable vector space with the trivial computable automorphism group, but with the dependence relations as complicated as possible, measured by their Turing degrees. As a corollary, we answer a question asked by A.S. Morozov in [Rigid constructive modules, Algebra and Logic, 28 (5) (1989) 570–583 (in Russian); (1990) 379–387 (English transl.)].